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oai:figshare.com:article/34036343 2026-09-30T17:50:11Z category_12 category_24 category_272 category_39 category_734 category_811 category_106 category_132 portal_5 item_type_1 month_year_09_2026

Examples of -bidegreed graph families (a-f) and the size normalized sensitivity of their neutral fixation probabilities to , shown separately for each initial mutant location and evaluated at (g-j). David A. Brewster (17118328) Yichen Huang (5794106) Michael Mitzenmacher (25145498) Martin A. Nowak (24845) Cell Biology Evolutionary Biology Environmental Sciences not elsewhere classified Ecology Biological Sciences not elsewhere classified Mathematical Sciences not elsewhere classified Science Policy Infectious Diseases unweighted undirected graphs edges denote interactions graph — compete evolutionary graph theory study mixed updating randomly chosen neighbor prove exact formulas div >< p study fixation probabilities short fixation times mixed updating graph represent fixation probabilities neighbors — evolutionary dynamics vacancy proportional two scenarios time step reproduction proportional population structure offspring replaces neutral evolution n </ efficient algorithm constant selection complex structures competitive replacement chosen uniformly adjacent vertex Vertices with degree d1 are represented as circles whereas vertices with degree are represented as squares. For the examples depicted, d2 is strictly greater than d1, but it is also possible for per Definition 3.1. Let n1 denote the number of vertices with degree d1 and n2 denote the number of vertices with degree d2, and set and . The value is the fixation probability of a neutral mutant starting at a vertex u of degree , its derivative with respect to is denoted , and the sensitivity is . Panels g–j display the size normalized sensitivity evaluated at and . Normalizing by N makes the sensitivity a function of alone, since by Theorem S3 in the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1014829#pcbi.1014829.s001" target="_blank">S1 Appendix</a> the sensitivity scales as 1/N at fixed . A graph of size N sitting at a plotted point therefore has sensitivity equal to the plotted value divided by N. a, a star graph, . The n2 = 1 center vertex has degree d2 = 6 while the n1 = 6 periphery vertices have degree d1 = 1 for a total of vertices. b, a path graph, . The intermediate vertices have degree 2 while the endpoint vertices have degree 1. c, a bipartite biregular graph; the vertices on the left side have degree d2 = 3 while the vertices on the right side have degree d1 = 2, for a total of N = 10 vertices. d, a fan graph . The center vertex has degree 2b = 6, while each of the 2 vertices on each of the b = 3 blades has degree 2, for a total of N = 2b + 1 = 7 vertices. e, a wheel graph . The center vertex has degree n = 8 while the n vertices on the spoke ends have degree 3, for a total of N = n + 1 = 9 vertices. f, a book graph . Each of the p = 3 pages has 2 page corners with degree 2, and the 2 seam endpoints have degree p + 1, for a total of N = 2p + 2 = 8 vertices. g, A contour plot in the space, where both parameters are dimensionless. The lines with shapes denote the values of and for the various bidegreed graphs in a-f. Note that the path graph has n1 = 2 and , so for . Thus as N increases, decreases. The intensity of the colorbar denotes , h, , i, , and j, . Lighter colors indicate a larger sensitivity. 2026-09-30T18:07:02Z Image Figure 10.1371/journal.pcbi.1014829.g003 https://figshare.com/articles/figure/_p_Examples_of_-bidegreed_graph_families_a-f_and_the_size_normalized_sensitivity_of_their_neutral_fixation_probabilities_to_shown_separately_for_each_initial_mutant_location_and_evaluated_at_g-j_p_/34036343 CC BY 4.0